Georgia Institute of Technology College of Sciences

One of the biggest hurdles in high performance computing today is the analysis of massive quantities of data. As the size of the datasets grows to petascale (and beyond), new techniques are needed to efficiently compute meaningful information from the raw data. Graph-based data (which is ubiquitous in social networks, biological interaction networks, etc) poses additional challenges due to the difficulty of parallelizing many common graph algorithms. A key component in success is the generation of "realistic" random data sets for testing and benchmarking new algorithms. The R-MAT graph generator introduced by Chakrabarti, Faloutsos, and Zhan (2004) offers a simple, fast method for generating very large directed graphs. One commonly held belief regarding graphs produced by R-MAT is that they are "scale free"; in other words, their degree distribution follows a power law as is observed in many real world networks. These properties have made R-MAT a popular choice for generating graphs for use in a variety of research disciplines including graph theoretic benchmarks, social network analysis, computational biology, and network monitoring. However, despite its wide usage and elegant, parsimonius design, our recent work provides the first rigorous mathematical analysis of the degree distributions of the generated graphs. Applying results from occupancy problems in probability theory, we derive exact expressions for the degree distributions and other parameters. We also prove that in the limit (as the number of vertices tends to infinity), graphs generated with R-MAT have degree distributions that can be expressed as a mixture of normal distributions. This talk will focus on the techniques used in solving this applied problem in terms of classical "ball and urn" results, including a minor extension of Chistyakov's theorem.

Graph tiling problems can be summarized as follows: given a graph H, what conditions do we need to find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. The most familiar example of a graph tiling problem is finding a matching in a graph. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove a degree condition that guarantees an arbitrary bipartite graph G will be tiled by an arbitrary bipartite graph H. We also prove this degree condition is best possible up to a constant. This answers a question of Zhao and proves an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.

In this talk I will discuss a new technique discovered by myself and Olof Sisask which produces many new insights in additive combinatorics, not to mention new proofs of classical theorems previously proved only using harmonic analysis. Among these new proofs is one for Roth's theorem on three-term arithmetic progressions, which gives the best bounds so far achieved by any combinatorial method. And another is a new proof that positive density subsets of the integers mod p contain very long arithmetic progressions, first proved by Bourgain, and improved upon by Ben Green and Tom Sanders. If time permits, I will discuss how the method can be applied to the 2D corners problem.